Spaces of phylogenetic networks Jonathan Klawitter PhD Exam 5th - - PowerPoint PPT Presentation

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Spaces of phylogenetic networks Jonathan Klawitter PhD Exam 5th - - PowerPoint PPT Presentation

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1 Spaces of phylogenetic networks Jonathan Klawitter PhD Exam 5th March, 2020 2 - 1 Phylogenetic trees & networks 2 - 2 Phylogenetic trees & networks 2 - 3 Phylogenetic trees & networks T 1 2 3 4 5 6 2 - 4

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1

Spaces of phylogenetic networks

Jonathan Klawitter PhD Exam · 5th March, 2020

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SLIDE 2

2 - 1

Phylogenetic trees & networks

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2 - 2

Phylogenetic trees & networks

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2 - 3

Phylogenetic trees & networks

1 2 3 4 5 6 T

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2 - 4

Phylogenetic trees & networks

1 2 3 4 5 6 T 1 2 3 4 5 6 N

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3 - 1

Set of networks

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3 1 2 3 4 1 4 3 2 1 3 2 4

. . .

1 4 2 3 1 4 3 2 1 3 2 4 1 3 2 4 1 2 1 2 1 3 2 4 1 3 2 4 4 3 4 3 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4

. . . . . .

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3 - 2

Set of networks

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3 1 2 3 4 1 4 3 2 1 3 2 4

. . .

1 4 2 3 1 4 3 2 1 3 2 4 1 3 2 4 1 2 1 2 1 3 2 4 1 3 2 4 4 3 4 3 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4

. . . trees . . .

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3 - 3

Set of networks

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3 1 2 3 4 1 4 3 2 1 3 2 4

. . .

1 4 2 3 1 4 3 2 1 3 2 4 1 3 2 4 1 2 1 2 1 3 2 4 1 3 2 4 4 3 4 3 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4

. . . trees normal . . .

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3 - 4

Set of networks

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3 1 2 3 4 1 4 3 2 1 3 2 4

. . .

1 4 2 3 1 4 3 2 1 3 2 4 1 3 2 4 1 2 1 2 1 3 2 4 1 3 2 4 4 3 4 3 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4

. . . trees normal . . . tree- child

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3 - 5

Set of networks

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 3 2 4 1 4 2 3 1 2 3 4 1 4 3 2 1 3 2 4

. . .

1 4 2 3 1 4 3 2 1 3 2 4 1 3 2 4 1 2 1 2 1 3 2 4 1 3 2 4 4 3 4 3 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4

. . . trees normal . . . tree- child tree-based . . .

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4 - 1

Rearrangement operations

NNI z x y y x z x y x y u u . . . x y x y

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4 - 2

Rearrangement operations

NNI z x y y x z x y x y u u SNPR 3 1 2 4 1 2 3 4 . . . x y x y

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4 - 3

Rearrangement operations

NNI z x y y x z x y x y u u SNPR 3 1 2 4 1 2 3 4 1 2 3 4 1 2 3 4 . . . x y x y

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4 - 4

Rearrangement operations

NNI z x y y x z x y x y u u SNPR 3 1 2 4 1 2 3 4 PR 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 . . . x y x y

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NNI tree space

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

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6 - 1

Chapters

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6 - 2

Chapters

Connectedness & diameter

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6 - 3

Chapters

Neighbourhood size Connectedness & diameter

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6 - 4

Chapters

Neighbourhood size Connectedness & diameter SNPR- and PR-distance

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6 - 5

Chapters

Neighbourhood size Connectedness & diameter SNPR- and PR-distance Maximum agreement graphs

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7 - 1

Connectedness & diameter

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7 - 2

Connectedness & diameter

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7 - 3

Connectedness & diameter

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8 - 1

Neighbourhood

N 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1

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8 - 2

Neighbourhood

Methods

Count number of possible operations Subtract number of trivial operations Correct for double counting

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8 - 3

Neighbourhood

Methods

Count number of possible operations Subtract number of trivial operations Correct for double counting

Results

Neighbourhood size for

trees under NNI and SNPR tree-child networks under NNI and SNPR normal networks under SNPR general networks (bounds)

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9 - 1

PR-distance

  • Theorem. Computing the SPR-distance of two trees is

NP-hard.

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9 - 2

PR-distance

  • Theorem. Computing the SPR-distance of two trees is

NP-hard.

  • Theorem. The space of trees embeds isometrically into the

space of networks unter SNPR and PR.

  • Corollary. Computing the SNPR- and PR-distance of two

networks is NP-hard.

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9 - 3

PR-distance

  • Theorem. Computing the SPR-distance of two trees is

NP-hard.

  • Theorem. The space of trees embeds isometrically into the

space of networks unter SNPR and PR.

  • Corollary. Computing the SNPR- and PR-distance of two

networks is NP-hard.

  • Theorem. Let N have r reticulations.

dPR(T, N) = min

T′∈D(N) dPR(T, T′) + r

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9 - 4

PR-distance

  • Theorem. Computing the SPR-distance of two trees is

NP-hard.

  • Theorem. The space of trees embeds isometrically into the

space of networks unter SNPR and PR.

  • Corollary. Computing the SNPR- and PR-distance of two

networks is NP-hard.

  • Theorem. Let N have r reticulations.

dPR(T, N) = min

T′∈D(N) dPR(T, T′) + r

  • Corollary. Computing dPR(T, N) is fixed-parameter

tractable.

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10 - 1

Maximum agreement forests

1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

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10 - 2

Maximum agreement forests

1 2 3 4 5 ρ 1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

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10 - 3

Maximum agreement forests

1 2 3 4 5 1 2 3 4 5 ρ ρ 1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

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10 - 4

Maximum agreement forests

1 2 3 4 5 ρ F 1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

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10 - 5

Maximum agreement forests

1 2 3 4 5 ρ F 1 2 3 4 5 ρ 1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

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10 - 6

Maximum agreement forests

1 2 3 4 5 ρ F 1 2 3 4 5 ρ ρ 1 2 3 4 5 1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

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10 - 7

Maximum agreement forests

1 2 3 4 5 ρ F 1 2 3 4 5 ρ ρ 1 2 3 4 5 1 2 3 4 5 T 1 2 3 4 5 T′ ρ 1 2 3 4 5 ρ ρ

  • Theorem. The size of a maximum agreement forest for

two trees characterises their SPR-distance.

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11 - 1

Maximum agreement graphs

1 2 3 4 5 N N′ ρ 1 2 3 4 5 ρ ρ 1 2 3 4 5

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11 - 2

Maximum agreement graphs

1 2 3 4 5 N N′ ρ 1 2 3 4 5 ρ ρ 1 2 3 4 5 1 2 3 4 5 ρ

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11 - 3

Maximum agreement graphs

1 2 3 4 5 N N′ ρ 1 2 3 4 5 ρ ρ 1 2 3 4 5 1 2 3 4 5 ρ G

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11 - 4

Maximum agreement graphs

1 2 3 4 5 N N′ ρ 1 2 3 4 5 ρ ρ 1 2 3 4 5 1 2 3 4 5 ρ G ρ 1 2 3 4 5

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11 - 5

Maximum agreement graphs

1 2 3 4 5 N N′ ρ 1 2 3 4 5 ρ ρ 1 2 3 4 5 1 2 3 4 5 ρ G ρ 1 2 3 4 5 ρ 1 2 3 4 5

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12 - 1

Agreement distance

  • Theorem. The number of sprouts and disagreement edges
  • f a maximum agreement graph define a

metric, the agreement distance dAD.

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12 - 2

Agreement distance

  • Theorem. The number of sprouts and disagreement edges
  • f a maximum agreement graph define a

metric, the agreement distance dAD.

  • Theorem. dAD(T, N) = dPR(T, N)
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12 - 3

Agreement distance

  • Theorem. The number of sprouts and disagreement edges
  • f a maximum agreement graph define a

metric, the agreement distance dAD.

  • Theorem. dAD(N, N′) ≤ dPR(N, N′) ≤ 3 dAD(N, N′)

dAD(N, N′) ≤ dSNPR(N, N′) ≤ 6 dAD(N, N′)

  • Theorem. dAD(T, N) = dPR(T, N)
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13 - 1

Publications

K., “The SNPR-Neighbourhood of tree-child networks”,

Journal of Graph Algorithms & Applications, 2018.

K., Linz, “On the Subnet Prune and Regraft Distance”,

Electronic Journal of Combinatorics, 2019.

K., “The agreement distance of rooted phylogenetic

networks ”, Discrete Mathematics & Theoretical Computer Science, 2019.

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13 - 2

Publications

K., “The SNPR-Neighbourhood of tree-child networks”,

Journal of Graph Algorithms & Applications, 2018.

K., Linz, “On the Subnet Prune and Regraft Distance”,

Electronic Journal of Combinatorics, 2019.

K., “The agreement distance of rooted phylogenetic

networks ”, Discrete Mathematics & Theoretical Computer Science, 2019.

Janssen, K., “Rearrangement operations on unrooted

phylogenetic networks ”, Theory and Applications of Graphs, 2019.

K., “The agreement distance of unrooted phylogenetic

networks ”, Discrete Mathematics & Theoretical Computer Science, 2020 (to appear).

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